Linear spaces on the intersection of cubic hypersurfaces
type: Item
instance of: scholarly article
title: Linear spaces on the intersection of cubic hypersurfaces
zbMATH Open document ID: 0546.10018
DOI: 10.1007/BF02349626
author: D. J. Lewis
author: Rainer Schulze-Pillot
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178180
review text: Let \(\lambda\) (r,m) denote the least n such that, for any prime p, any set of r cubic forms in n variables over \({\mathbb{Q}}_ p\) has an m- dimensional linear space of common p-adic solutions. Similarly let \(\Lambda\) (r,m) denote the corresponding quantity when \({\mathbb{Q}}_ p\) is replaced by \({\mathbb{Q}}\). The existence of \(\lambda\) (r,m) was established by \textit{R. Brauer} [Bull. Am. Math. Soc. 51, 749-755 (1945)] and that of \(\Lambda\) (r,m) by \textit{B. J. Birch} [Mathematika 4, 102-105 (1957; Zbl 0081.045)]. Indeed in each case a far more general result was obtained. While the works of Brauer and Birch are effective, the estimates for \(\lambda\) (r,m) and \(\Lambda\) (r,m) that one would obtain by following their methods would be extremely large. The present paper gives the bounds \(\lambda (r,m)\ll r^ 2m^ 2+r^ 4m\) for r,\(m\geq 1\), and \(\Lambda (1,m)\ll m^{\alpha}\) for \(m\geq 1\), where \(\alpha ={1\over2}(5+\sqrt{17})=4.56... .\) Moreover it is shown that \(\Lambda (r,m)\ll r^{11}m+r^ 3m^ 5\) and that \(\Lambda (r,m)\ll r^ 5m^{14}\) (r,\(m\geq 1\) in either case). The method is based on elementary algebraic geometry, the only analytic ingredient being an estimate for \(\lambda\) (r,1) due to \textit{W. M. Schmidt} [Monatsh. Math. 93, 211-223 (1982; Zbl 0473.10017)].
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 3869428
zbMATH Keywords: upper bounds for number of variables
zbMATH Keywords: existence of m-dimensional linear variety
zbMATH Keywords: common zero set
zbMATH Keywords: Brauer induction
zbMATH Keywords: forms in many variables
zbMATH Keywords: cubic forms
zbMATH Keywords: common p-adic solutions
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
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OpenAlex ID: W2083109249
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description: scientific article; zbMATH DE number 3869428